Some Non-Normal Cayley Digraphs of the Generalized Quaternion Group of Certain Orders

We show that an action of SL(2, p), p ≥ 7 an odd prime such that 4 6 | (p − 1), has exactly two orbital digraphs Γ1, Γ2, such that Aut(Γi) admits a complete block system B of p + 1 blocks of size 2, i = 1, 2, with the following properties: the action of Aut(Γi) on the blocks of B is nonsolvable, doubly-transitive, but not a symmetric group, and the subgroup of Aut(Γi) that fixes each block of B set-wise is semiregular of order 2. If p = 2k − 1 > 7 is a Mersenne prime, these digraphs are also Cayley digraphs of the generalized quaternion group of order 2k+1. In this case, these digraphs are non-normal Cayley digraphs of the generalized quaternion group of order 2k+1. There are a variety of problems on vertex-transitive digraphs where a natural approach is to proceed by induction on the number of (not necessarily distinct) prime factors of the order of the graph. For example, the Cayley isomorphism problem (see [6]) is one such problem, as well as determining the full automorphism group of a vertex-transitive digraph Γ. Many such arguments begin by finding a complete block system B of Aut(Γ). Ideally, one would then apply the induction hypothesis to the groups Aut(Γ)/B and fixAut(Γ)(B)|B, where Aut(Γ)/B is the permutation group induced by the action of Aut(Γ) on B, and fixAut(Γ)(B) is the subgroup of Aut(Γ) that fixes each block of B set-wise, and B ∈ B. Unfortunately, neither Aut(Γ)/B nor fixAut(Γ)(B)|B need be the automorphism group of a digraph. In fact, there are examples of vertex-transitive graphs where Aut(Γ)/B is a doubly-transitive nonsolvable group that is not a symmetric group (see [7]), as well as examples of vertex-transitive graphs where fixAut(Γ)(B)|B is a doubly-transitive nonsolvable group that is not a symmetric group (see [2]). (There are also examples where Aut(Γ)/B is a solvable doubly-transitive group, but in practice, this is not usually the electronic journal of combinatorics 10 (2003), #R31 1 a genuine obstacle in proceeding by induction.) The only known class of examples of vertex-transitive graphs where Aut(Γ)/B is a doubly-transitive nonsolvable group, have the property that Aut(Γ)/B is a faithful representation of Aut(Γ) and Γ is not a Cayley graph. In this paper, we give examples of vertex-transitive digraphs that are Cayley digraphs and the action of Aut(Γ)/B on B is doubly-transitive, nonsolvable, not faithful, and not a symmetric group.